On the asymptotic behavior of Kac-Wakimoto characters

Abstract:

Recently, Kac and Wakimoto established specialized character formulas for irreducible highest weight $s\ell (m,1)^\wedge$ modules and established a main exponential term in their asymptotic expansions. By different methods, we improve upon the Kac-Wakimoto asymptotics for these characters, obtaining an asymptotic expansion with an arbitrarily large number of terms beyond the main term. More specifically, it is well known that in the case of holomorphic modular forms, asymptotic information may be obtained using modular transformation properties. However, here this is not the case due to the analytic nature of the Kac-Wakimoto series as discovered recently by the first author and Ono. We first “complete” these series by adding to them certain integrals, obtaining functions that exhibit suitable modular transformation laws, at the expense of the completed objects being nonholomorphic. We then exploit this mock-modular behavior of the Kac-Wakimoto series to obtain our asymptotic expansion. In particular, we show that beyond the main term, the asymptotic behavior is dictated by the nonholomorphic part of the completed Kac-Wakimoto characters, which is a priori invisible. Euler numbers (equivalently, zeta-values) appear as coefficients.